91Ó°ÊÓ

To find the range of \(\theta\), we need to consider the hypocycloid in the first quadrant, which implies that \(\cos{\theta}\) and \(\sin{\theta}\) should both be between \(0\) and \(1\). Since the powers are odd, the ranges for both of them will still be the same. The respective equation can be written as \[x = a(\cos^3{\theta})\quad and \quad y = a(\sin^3{\theta})\] Since both \(x\) and \(y\) are in the first quadrant, the range of \(\theta\) must be \(0 \le \theta \le \frac{\pi}{2}\). ##Step 2: Express y in terms of x##

In order to use the disk method, we need to express \(y\) in terms of \(x\). From the equation \(x = a(\cos^3{\theta})\), we can write \[\cos^3{\theta} = \frac{x}{a}\] Taking the cube root on both sides, \[\cos{\theta} = \left(\frac{x}{a}\right)^{1/3}\] Now, rewrite the equation for \(y\) using the relation between \(\cos{\theta}\) and \(\sin{\theta}\), which is \(\sin^2\theta + \cos^2\theta = 1\) \[y = a\left(\sin^3{\theta}\right) = a \left(\sin^2{\theta}\right)^{3/2}\] Using the relation \(\sin^2\theta = 1 - \cos^2\theta\), \[y = a\left(1-\cos^2\theta\right)^{3/2}\] Substituting the value of \(\cos{\theta}\) from the previous step, \[y = a\left(1 - \left(\frac{x}{a}\right)^{\frac{2}{3}}\right)^{\frac{3}{2}}\] ##Step 3: Set up and evaluate the integral##

Now that we have expressed \(y\) in terms of \(x\), we can set up the integral for calculating the volume using the disk method: \[V = \pi \int_0^a \left(a\left(1-\left(\frac{x}{a}\right)^{\frac{2}{3}}\right)^{\frac{3}{2}}\right)^2 dx\] This integral can be simplified to: \[V = \pi a^2 \int_0^a \left(1 - \left(\frac{x}{a}\right)^{\frac{2}{3}}\right)^3 dx\] To solve the integral, let \(u = 1 - \left(\frac{x}{a}\right)^{\frac{2}{3}}\). Then \(\frac{du}{-2 / 3a} = dx\). The integral becomes: \[V = -\frac{3}{2}\pi a ^2 \int_{1}^{0} u^3 \, du\] Calculating the integral: \[V = -\frac{3}{2}\pi a^2 \left[\frac{1}{4}u^4\Big{|}_1^0\right]\] \[V = -\frac{3}{2}\pi a^2 \left[0 - \frac{1}{4}\right]\] \[V = \frac{3}{8}\pi a^2\] The volume of the solid generated by revolving the given hypocycloid around the X-axis is \(\frac{3}{8}\pi a^2\).